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On the roles of ``stability'' and ``convergence'' in semidiscrete projection methods for initial-value problems

Author: Seymour V. Parter
Journal: Math. Comp. 34 (1980), 127-154
MSC: Primary 65N35; Secondary 34G99
MathSciNet review: 551294
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Abstract: Consider the initial value problem (1.1)

$\displaystyle \frac{d}{{dt}}u(t) = Au(t) + f(t),\quad t > 0,$


$\displaystyle u(0) = {u_0},$

where A is a linear operator taking $ D(A) \subset X$ into X, where X is a Banach space. Consider also semidiscrete numerical methods of the form: find $ {U_N}(t):[0,T] \to {X_N}$ such that $ (1.1\prime )$

$\displaystyle \frac{{d{U_N}}}{{dt}} = {A_N}{U_N} + {P_N}f,$

$ (1.2\prime )$

$\displaystyle {U_N}(0) = U_N^0 \in {X_N},$

where $ {X_N}$ is a finite dimensional subspace and $ {P_N}$ is a projector onto $ {X_N}$.

The study of such numerical methods may be related to the approximation of semigroups and Laplace transform methods making use of the resolvent operators $ {(A - \lambda I)^{ - 1}},{({A_N} - \lambda {I_N})^{ - 1}}$. The basic results require stability or weak stability and give convergence rates of the same order as in the steady state problems.

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Article copyright: © Copyright 1980 American Mathematical Society